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namely for a surface as we ordinarily imagine surfaces. It is
possible from the formula

expressing (as before) the square of the distance between neigh-
bouring points on a given surface (determined when the func-
tions givgiz>gzz are given), to calculate the measure of curvature
of any point of the surface wholly in terms of the given functions
gn.g12sg22' ^<^> in ordinary language, to speak of the 'curva-
ture' of a space of more than t&o dimensions is to make a
meaningless noise. Nevertheless Riemann, generalizing Gauss,
proceeded in the same mathematical way to build up an expres-
sion involving all the g's in the general case of an ^-dimensional
space, which is of the same kind mathematically as the Gaussian
expression for the curvature of a surface, and this generalized
expression is what he called the measure of curvature of the
space. It is possible to exhibit visual representations of a curved
space of more than two dimensions, but such aids to perception
are about as useful as a pair of broken crutches to a ma** with
no feet, for they add nothing to the understanding and they are
mathematically useless.
Why did Riemann do all this and what has come out of it?
Not attempting to answer the first, except to suggest that
Riemann did what he did because his daemon drove him, we
may briefly enumerate some of the gains that have accrued
from Riemann'1 s revolution in geometrical thought. First, it put
the creation of 'spaces* and 'geometries* in unlimited number
for specific purposes  use in dynamics, or hi pure geometry, or
in physical science - within the capabilities of professional geo-
meters, and it baled together huge masses of important geo*
metrical theorems into compact bundles that could be handled
easily as wholes. Second, it clarified our conception of space, at
least so far as mathematicians deal in 'space*, and stripped that
mystic nonentity Space of its last shred of mystery. Riemann's
achievement has taught mathematicians to disbelieve in any
geometry, or in any space, as a necessary mode of human per-
ception. It was the last nail in the coffin of absolute space, and
the first in thai of the 'absolutes* of nineteenth-century physics.